Problem 102 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 8(-4)+5(-4)-6 $$

Step-by-Step Solution

Answer

-58

Step 1: Apply Distribution

Start by applying the coefficients to each value within the expression: $$8(-4)$$ and $$5(-4)$$.

Step 2: Multiply the Terms

Multiply the coefficients by the terms inside the parentheses: $$8 \times (-4) = -32$$ and $$5 \times (-4) = -20$$.

Step 3: Rewrite the Expression

After the multiplication step, rewrite the expression with the new values: $$-32 - 20 - 6$$.

Step 4: Combine Like Terms

Combine all the terms in the expression: $$-32 - 20 - 6 = -58.$$

Key Concepts

Distribution in Algebra

In algebra, distribution refers to the process of multiplying each term inside a parenthesis by the factor outside of it. This rule is called the distributive property and is written as: \[a(b + c) = ab + ac\].

For instance, if you have an expression like \(8(-4) + 5(-4) - 6\), you need to distribute the coefficients (8 and 5) to the terms inside the parentheses (-4).

This means you’ll first multiply 8 by -4, then 5 by -4. This step helps in simplifying the expression by removing the parentheses, making the equation easier to solve.

Multiplication of Integers

Multiplication of integers involves combining positive and negative numbers according to simple rules.

Here are some key points to remember:

• Multiplying two positive numbers or two negative numbers results in a positive product: i.e.,
  \[4 \times 3 = 12\] and \[-4 \times (-3) = 12\].
  
• Multiplying a positive number by a negative number results in a negative product: i.e., \[4 \times (-3) = -12\].

Let's apply these to our exercise: when you multiply \(8 \times -4\), you get \(-32\). Similarly, \(5 \times -4\) becomes \(-20\).

Understanding these rules makes distributing and simplifying expressions much more straightforward.

Combining Like Terms

Combining like terms is a fundamental skill in algebra that simplifies expressions. Like terms are terms that contain the same variable, raised to the same power, or constant terms without variables.

For example, in the expression \(-32 - 20 - 6\), all terms are constants (they don't have any variables).
To combine them, you simply sum their coefficients.

\[-32 - 20 - 6\] can be simplified step-by-step:

• First, combine \(-32 - 20\), getting \(-52\).
• Then, subtract 6 from \(-52\), resulting in \(-58\).

After combining like terms, the expression is simplified to its final form. This makes it easier to understand the value represented by the expression.